We’re taking very basic differential equations in school and I know how to solve them (more or less) but I think I’d be better when I u derstand what this is doing. Like we use differentiation to find the slope of a curve integration to find the curve equation from a given slope of a point or find the area under the curve
When I solve a differential equation. What is it that I do?
In: Mathematics
Let’s imagine a big empty field, surrounded by walls. We plant some grass seed all over the field. Each day, the grass will grow at some rate. The rate isn’t constant, or the grass would eventually be so tall that it would be up in space. So the rate of change of the amount of grass is dependent on how much grass there is already.
Now let’s put some mice in the field. The mice eat grass, and if they have enough grass to eat, they will live long enough to reproduce and make more mice. At first, with a small number of mice and plenty of grass, the mice will reproduce a lot and there will still be a lot of grass. But as time goes on, eventually the rate that the grass grows each day won’t be enough to keep up with the number of new mice. Some of the mice will starve to death and not reproduce.
So now for the grass, its rate of change each day depends on how much grass there is already and how many mice are eating it.
And for the mice, the change in their population each day depends both on how many mice there are (more reproduction) and how much grass there is (less reproduction).
Finally, let’s add some cats. The cats only eat mice, and they need quite a lot of mice in order to live long enough to reproduce. Just like the mice with the grass, the cats will reproduce more if there are lots of mice, and less when there aren’t enough mice.
It shouldn’t come as a surprise that we now have a couple more relationships: the change in the number of mice now depends on how many cats there are (cats eat mice), and the number of new cats depends on how many mice there are (cats can die from starvation before they can reproduce).
All of these relationships involve both *amounts* and *rates of change*. Let’s look at all of the dependencies we’ve found:
Grass rate of change: how much grass, how many mice
Mice rate of change: how much grass, how many mice, how many cats
Cats rate of change: how many cats, how many mice
When we write out all of these relationships with specific mathematical values, we get a “system of differential equations”. On its own, this system isn’t super useful. It just tells us that if we know all the specific amounts for a given day, we can predict the amounts on the next day.
However, if we could “solve” this system of equations by turning it into a single formula for how much grass there is over time, one for how many mice there are over time, and one for how many cats there are over time, we can use the initial amounts to reason about what the system will look like at any date in the future, as well as being able to draw graphs of each population to reason about the overall dynamic behavior of the system.
In this case, we may see that each population experiences periodic rises and falls, kind of like a sine wave (but shaped a bit different). For example, when there aren’t many mice, the cats die off and the grass can grow a lot. Fewer cats and more grass are perfect conditions for mice to reproduce quickly, so they will! But eventually, there will be about as many mice as the grass can support, and the cats will start being well-fed, so the cat population will rise. Lots of cats and not much grass is really bad for mice… their numbers will go down now. The other populations (grass and cats) will have similar ups and downs.
There may be an “equilibrium” – an amount of grass, mice, and cats that is somehow “perfect” such that all of the change rates are zero. If we start at the equilibrium levels, we will just stay there forever. But if we start somewhere else, we might see the populations get closer and closer to the equilibrium, or oscillate around it (basically trying to get there but always overshooting it because the other populations aren’t there at the same time), or it may do both (oscillate but get closer to equilibrium over time).
Much of what you learn in differential equations is designed to help you reason about systems like this: dynamic systems where *quantities* and *rates of change* have dependencies which are known, and we want to reason about the long-term behavior of the system.
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